We investigate a rich new class of exactly solvable particle systems generalizing the Totally Asymmetric Simple Exclusion Process (TASEP). Our particle systems evolve in discrete or continuous space and can be thought of as new exactly solvable examples of tandem queues, directed first- or last-passage percolation models, or Robinson-Schensted-Knuth type systems with random input. One of the features of the particle systems we consider is the presence of spatial inhomogeneity which can lead to the formation of traffic jams.
For systems with special step-like initial data, we find explicit limit shapes, describe their hydrodynamic evolution, and obtain asymptotic fluctuation results which put our generalized TASEPs into the Kardar-Parisi-Zhang universality class. At a critical scaling around a traffic jam in the continuous space TASEP, we observe deformations of the Tracy-Widom distribution and the extended Airy kernel, revealing the finer structure of this novel type of phase transitions.
A homogeneous version of our discrete space system is a one-parameter deformation of the geometric last-passage percolation (Johansson, 2000), and we obtain an extension of the last-passage percolation limit shape parabola.
The exact solvability and asymptotic behavior of generalizations of TASEP we study are powered by a new nontrivial connection to Schur measures and processes.